Efficient calculation of real-space lattice propagators in the presence of a Fermi sea

We present an efficient method for computing the real-space propagators (lattice Green's functions) of any tight-binding Hamiltonian in the presence of a Fermi sea with carrier concentration $x$. The method is valid for any lattice, dispersion, and dimension, provided the corresponding $x=0$ propagators are known. We show that suitable combinations of the finite-$x$ particle-addition propagators have a real or imaginary part trivially related to their $x=0$ counterpart, while the remaining part follows from a Kramers-Kronig relation that can be evaluated for all energies at once using the fast Fourier transforms. The computational cost is therefore independent of the dimensionality, unlike that of direct Brillouin-zone integration. The particle-removal propagators follow from general identities. We validate the method against direct integration for hypercubic lattices in one, two, and three dimensions, and use the 2D square lattice to illustrate how the shape of the Fermi surface is imprinted on the spatial structure of the propagators. In particular, at energies far outside the band, the propagator maps converge to the Fraunhofer diffraction pattern whose aperture is the unoccupied part of the Brillouin zone.